Complex calcium oscillations and the role of mitochondria and cytosolic proteins

Intracellular calcium oscillations, which are oscillatory changes of cytosolic calcium concentration in response to agonist stimulation, are experimentally well observed in various living cells. Simple calcium oscillations represent the most common pattern and many mathematical models have been published to describe this type of oscillation. On the other hand, relatively few theoretical studies have been proposed to give an explanation of complex intracellular calcium oscillations, such as bursting and chaos. In this paper, we develop a new possible mechanism for complex calcium oscillations based on the interplay between three calcium stores in the cell: the endoplasmic reticulum (ER), mitochondria and cytosolic proteins. The majority ( approximately 80%) of calcium released from the ER is first very quickly sequestered by mitochondria. Afterwards, a much slower release of calcium from the mitochondria serves as the calcium supply for the intermediate calcium exchanges between the ER and the cytosolic proteins causing bursting calcium oscillations. Depending on the permeability of the ER channels and on the kinetic properties of calcium binding to the cytosolic proteins, different patterns of complex calcium oscillations appear. With our model, we are able to explain simple calcium oscillations, bursting and chaos. Chaos is also observed for calcium oscillations in the bursting mode.     

Cytosolic calcium oscillators

Many cells display oscillations in intracellular calcium resulting from the periodic release of calcium from intracellular reservoirs. Frequencies are varied, but most oscillations have periods ranging from 5 to 60 s. For any given cell, frequency can vary depending on external conditions, particularly the concentration of natural stimuli or calcium. This cytosolic calcium oscillator is particularly sensitive to those stimuli (neurotransmitters, hormones, growth factors) that hydrolyze phosphoinositides to give diacylglycerol and inositol 1,4,5-trisphosphate (Ins1,4,5P3). The ability of Ins1,4,5P3 to mobilize intracellular calcium is a significant feature of many of the proposed models that are used to explain oscillatory activity. Receptor-controlled oscillator models propose that there are complex feedback mechanisms that generate oscillations in the level of Ins1,4,5P3. Second messenger-controlled oscillator models demonstrate that the oscillator is a component of the calcium reservoir, which is induced to release calcium by a constant input of either Ins1,4,5P3 or calcium itself. In the latter case, the process of calcium-induced calcium release might be the basis of oscillatory activity in many cell types. The function of calcium oscillations is still unknown. Because oscillator frequency can vary with agonist concentration, calcium transients might be part of a frequency-encoded signaling system. When an external stimulus arrives at the cell surface the information is translated into a train of calcium spikes, i.e., the signal is digitized. Certain cells may then convey information by varying the frequency of this digital signal.     

极低频磁场对激动剂诱发钙振荡的影响

从激动剂诱发钙振荡的非线性动力学模型出发, 通过数值计算分析极低频磁场对胞内游离钙离子浓度［Ｃａ２ ＋ ］ｉ 的影响。研究结果表明：只有当外加磁场的频率与胞内钙振荡的特征频率相近时,极低频磁场才会对该细胞的［Ｃａ２ ＋］ｉ 产生影响;由于激动剂诱发钙振荡的动力学模型中的许多参数是因细胞而异的,因此极低频磁场对［Ｃａ２ ＋ ］ｉ 的影响具有显著的个体差异     

A membrane model for cytosolic calcium oscillations. A study using Xenopus oocytes.

Cytosolic calcium oscillations occur in a wide variety of cells and are involved in different cellular functions. We describe these calcium oscillations by a mathematical model based on the putative electrophysiological properties of the endoplasmic reticulum (ER) membrane. The salient features of our membrane model are calcium-dependent calcium channels and calcium pumps in the ER membrane, constant entry of calcium into the cytosol, calcium dependent removal from the cytosol, and buffering by cytoplasmic calcium binding proteins. Numerical integration of the model allows us to study the fluctuations in the cytosolic calcium concentration, the ER membrane potential, and the concentration of free calcium binding sites on a calcium binding protein. The model demonstrates the physiological features necessary for calcium oscillations and suggests that the level of calcium flux into the cytosol controls the frequency and amplitude of oscillations. The model also suggests that the level of buffering affects the frequency and amplitude of the oscillations. The model is supported by experiments indirectly measuring cytosolic calcium by calcium-induced chloride currents in Xenopus oocytes as well as cytosolic calcium oscillations observed in other preparations.     

A mesoscopic stochastic mechanism of cytosolic calcium oscillations

Based on a model of intracellular calcium (Ca(2+)) oscillation with self-modulation of inositol 1,4,5-trisphosphate signal, the mesoscopic stochastic differential equations for the intracellular Ca(2+) oscillations are theoretically derived by using the chemical Langevin equation method. The effects of the finite biochemical reaction molecule number on both simple and complex cytosolic Ca(2+) oscillations are numerically studied. In the case of simple intracellular Ca(2+) oscillation, it is found that, with the increase of molecule number, the coherence resonance or autonomous resonance phenomena can occur for some external stimulation parameter values. In the cases of complex cytosolic Ca(2+) oscillations, each extremum of concentration of cytosolic Ca(2+) oscillations corresponds to a peak in the histogram of Ca(2+) concentration, and the most probability appeared during the bursting plateau level for bursting, but at the largest minimum of Ca(2+) concentration for chaos. For quasi-periodicity, however, there are only two peaks in the histogram of Ca(2+) concentration, and the most probability is located at low concentration state.     

Modelling receptor-controlled intracellular calcium oscillators.

This paper presents mathematical models for the hepatocyte calcium oscillator which follow the concepts in a class of informal models developed to account for the striking dependence on the receptor type of several features of the calcium oscillations, in particular the shape and duration of the free calcium transients. The essence of these models is that the transients should be timed by a build-up of activated GTP-binding proteins, which, combined with positive feedback processes and perhaps with cooperative effects, leads to a sudden activation of phospholipase C (PLC), followed by negative feedback processes which switch off the calcium rise and lead to a fall in free calcium back to resting levels. These models predict pulsatile oscillations in inositol (1,4,5)P3 as well as in free calcium. We show that receptor-controlled intracellular calcium oscillators involving an unknown positive feedback pathway onto PLC and negative feedback from protein kinase C (PKC) onto G-proteins and receptors, or negative feedback by stimulation of GTPase activity can simulate many of the features of observed intracellular calcium oscillations. These oscillators exhibit a dependence of frequency on agonist concentration and a dependence of transient duration on receptor and G-protein type. We also show that a PLC-dependent GTPase activating factor (GAF) could provide explanations for some otherwise puzzling features of intracellular calcium oscillations.     

Modeling of molecular and cellular mechanisms involved in Ca2+ signal encoding in airway myocytes

In airway myocytes signal transduction via cytosolic calcium plays an important role. In relation with experimental results we review models of basic molecular and cellular mechanisms involved in the signal transduction from the myocyte stimulation to the activation of the contractile apparatus. We concentrate on mechanisms for encoding of input signals into Ca²⁺ signals and the mechanisms for their decoding. The mechanisms are arranged into a general scheme of cellular signaling, the so-called bow-tie architecture of signaling, in which calcium plays the role of a common media for cellular signals and links the encoding and decoding part. The encoding of calcium signals in airway myocytes is better known and is presented in more detail. In particular, we focus on three recent models taking into account the intracellular calcium handling and ion fluxes through the plasma membrane. The model of membrane conductances was originally proposed for predicting membrane depolarization and voltage-dependent Ca²⁺ influx triggered by initial cytosolic Ca²⁺ increase as observed on cholinergic stimulation. Cellular models of intracellular Ca²⁺ handling were developed to investigate the role of a mixed population of InsP₃ receptor isoforms and the cellular environment in the occurrence of Ca²⁺ oscillations, and the respective role of the sarcoplasmic reticulum, mitochondria, and cytosolic Ca²⁺-binding proteins in cytosolic Ca²⁺ clearance. Modeling the mechanisms responsible for the decoding of calcium signals is developed in a lesser extent; however, the most recent theoretical studies are briefly presented in relation with the known experimental results.     

Calcium and cell cycle progression: possible effects of external perturbations on cell proliferation.

Exit from the phase of cellular division appears to be driven by a calcium signal that triggers a cascade of events leading to the completion of mitosis. Here we propose a model that relates the dynamics of cytosolic calcium to progression through mitosis, G1 and G2 phases of the cell cycle. To this end, the assumption has been made that the transient rise ir cytosolic calcium concentration during mitosis is induced by inositol(1,4,5)triphosphate (IP3), which in turn is released at high levels of mitosis-promoting factor (MPF). On this basis, a system of ordinary differential equations is proposed to simulate the evolution of ten cell-cycle-specific molecular species, including cyclins A and B, MPF, IP3, Ca2+, the CaMKII holoenzyme, and the ubiquitination complex. The influence on the cell proliferation capacity exerted by external perturbations, like calcium microinjections, depletion of intracellular calcium stores, electromagnetic fields, or stimulation/inhibition of different calcium currents through the plasma membrane, can be studied by appropriate modulation of the parameters involved in the signal transduction pathway.     

Stimulus-dependent control of inositol 1,4,5-trisphosphate-induced Ca(2+) oscillation frequency by the endoplasmic reticulum Ca(2+)-ATPase

In many cell types, receptor stimulation evokes cytosolic calcium oscillations with a frequency that depends on agonist dose. Previous studies demonstrated controversial effects of changing the activity of the endoplasmic reticulum Ca(2+)-ATPase upon the frequency of oscillations. By numerical simulations, we found that the model of De Young and Keizer (J. Keizer and G.W. De Young, 1994, J. Theor. Biol. 166: 431-442), unlike other models, can explain the observed discrepancies, assuming that the different experiments were performed at different stimulus levels. According to model predictions, partial inhibition of internal calcium pumps is expected to increase frequency at low stimulus strength and should have an opposite effect at strong stimuli. Similar results were obtained using an analytical estimation of oscillation period, based on calcium-dependent channel activation and inactivation. In experiments on HeLa cells, 4 nM thapsigargin increased the frequency of calcium oscillations induced by 1 and 2.5 microM histamine but had no effect on supramaximally stimulated cells. In HEp-2 cells, 2 nM thapsigargin slowed down the rapid, ATP-induced oscillations. Our results suggest that in the investigated cell types, the De Young-Keizer model based on inositol 1,4,5-trisphosphate-dependent calcium-induced calcium release can properly describe intracellular calcium oscillations.     

Model for external influences on cellular signal transduction pathways including cytosolic calcium oscillations

Experiments on the effects of extremely-low-frequency (ELF) electric and magnetic fields on cells of the immune system, T-lymphocytes in particular, suggest that the external field interacts with the cell at the level of intracellular signal transduction pathways. These are directly connected with changes in the calcium-signaling processes of the cell. Based on these findings, a theoretical model for receptor-controlled cytosolic calcium oscillations and for external influences on the signal transduction pathway is presented. We discuss the possibility that the external field acts on the kinetics of the signal transduction between the activated receptors at the cell membrane and the G-proteins. It is shown that, depending on the specific combination of cell internal biochemical and external physical parameters, entirely different responses of the cell can occur. We compare the effects of a coherent (periodic) modulation and of incoherent perturbations (noise). The model and the calculations are based on the theory of self-sustained, nonlinear oscillators. It is argued that these systems form an ideal basis for information-encoding processes in biological systems. © 1995 Wiley-Liss, Inc.     

Mechanical and biochemical modeling of cortical oscillations in spreading cells

Actomyosin-based cortical contractility is a common feature of eukaryotic cells and is involved in cell motility, cell division, and apoptosis. In nonmuscle cells, oscillations in contractility are induced by microtubule depolymerization during cell spreading. We developed an ordinary differential equation model to describe this behavior. The computational model includes 36 parameters. The values for all but two of the model parameters were taken from experimental measurements found in the literature. Using these values, we demonstrate that the model generates oscillatory behavior consistent with current experimental observations. The rhythmic behavior occurs because of the antagonistic effects of calcium-induced contractility and stretch-activated calcium channels. The model makes several experimentally testable predictions: 1), buffering intracellular calcium increases the period and decreases the amplitude of cortical oscillations; 2), increasing the number or activity of stretch activated channels leads to an increase in period and amplitude of cortical oscillations; 3), inhibiting Ca²⁺ pump activity increases the period and amplitude of oscillations; and 4), a threshold exists for the calcium concentration below which oscillations cease.     

Birhythmicity, trirhythmicity and chaos in bursting calcium oscillations

We have analyzed various types of complex calcium oscillations. The oscillations are explained with a model based on calcium-induced calcium release (CICR). In addition to the endoplasmic reticulum as the main intracellular Ca2+ store, mitochondrial and cytosolic Ca2+ binding proteins are also taken into account. This model was previously proposed for the study of the physiological role of mitochondria and the cytosolic proteins in gene rating complex Ca2+ oscillations [1]. Here, we investigated the occurrence of different types of Ca2+ oscillations obtained by the model, i.e. simple oscillations, bursting, and chaos. In a bifurcation diagram, we have shown that all these various modes of oscillatory behavior are obtained by a change of only one model parameter, which corresponds to the physiological variability of an agonist. Bursting oscillations were studied in more detail because they express birhythmicity, trirhythmicity and chaotic behavior. Two different routes to chaos are observed in the model: in addition to the usual period doubling cascade, we also show intermittency. For the characterization of the chaotic behavior, we made use of return maps and Lyapunov exponents. The potential biological role of chaos in intracellular signaling is discussed.     

Recovery of Ins(1,4,5)-trisphosphate-dependent calcium signaling in neonatal gonadotrophs

Pituitary gonadotrophs express non-desensitizing gonadotropin-releasing hormone (GnRH) receptors and their activations leads to inositol 1,4,5-trisphosphate (InsP3)-dependent Ca2+ mobilization. When added in physiological concentration range GnRH induces baseline Ca2+ oscillations, whereas in higher concentrations it induces a prolonged spike response accompanied with non-oscillatory or oscillatory plateau response. Here, we studied the recovery of calcium signaling during repetitive stimulation with short (10-30 s) GnRH pulses and variable interpulse intervals in neonatal gonadotrophs perfused with Ca2+/Na+ -containing, Ca2+ -deficient/Na+ -containing, and Ca2+ -containing/Na+ -deficient media. In Ca2+/Na+ -containing medium, baseline Ca2+ oscillations recovered without refractory period and with a time constant of approximately 20 s, whereas the recovery of spike response occurred after 25-35 s refractory period and with a time constant of approximately 30 s. During repetitive GnRH stimulation, removal of Ca2+ had only a minor effect on baseline oscillations but abolished spike response, whereas removal of Na+ slightly extended duration of baseline oscillations and considerably prolonged spike response. These results indicate that two calcium handling mechanisms are operative in gonadotrophs: redistribution of calcium within InsP3-sensitive and -insensitive pools and a sodium-dependent calcium efflux followed by calcium influx. Redistribution of Ca2+ within the cell leads to rapid recovery of InsP3-dependent pool, whereas the Na+ -dependent Ca2+ efflux pathway is activated by spike response and limits the time of exposure to elevated cytosolic Ca2+ concentrations.     

Mitochondria regulate the amplitude of simple and complex calcium oscillations.

In a mathematical model for simple calcium oscillations [Biophys. Chem. 71 (1998) 125], it has been shown that mitochondria play an important role in the maintenance of constant amplitudes of cytosolic Ca(2+) oscillations. Simple plausible rate laws for Ca(2+) fluxes across the inner mitochondrial membrane have been used in this model. Here we show that it is possible to use the same rate laws as a plug-in element in other existing mathematical models and obtain the same effect on amplitude regulation. This result appears to be universal, independent of the type of model and the type of Ca(2+) oscillations. We demonstrate this on two models for spiking Ca(2+) oscillations [J. Biol. Chem. 266 (1991) 11068; Cell Calcium 14 (1993) 311] and on two recent models for bursting Ca(2+) oscillations; one of them being a receptor-operated model [Biophys. J. 79 (2000) 1188] and the other one being a store-operated model [BioSystems 57 (2000) 75].     

Distinct External Signals Trigger Sequential Release of Apical Organelles during Erythrocyte Invasion by Malaria Parasites

The invasion of erythrocytes by Plasmodium merozoites requires specific interactions between host receptors and parasite ligands. Parasite proteins that bind erythrocyte receptors during invasion are localized in apical organelles called micronemes and rhoptries. The regulated secretion of microneme and rhoptry proteins to the merozoite surface to enable receptor binding is a critical step in the invasion process. The sequence of these secretion events and the external signals that trigger release are not known. We have used time-lapse video microscopy to study changes in intracellular calcium levels in Plasmodium falciparum merozoites during erythrocyte invasion. In addition, we have developed flow cytometry based methods to measure relative levels of cytosolic calcium and study surface expression of apical organelle proteins in P. falciparum merozoites in response to different external signals. We demonstrate that exposure of P. falciparum merozoites to low potassium ion concentrations as found in blood plasma leads to a rise in cytosolic calcium levels through a phospholipase C mediated pathway. Rise in cytosolic calcium triggers secretion of microneme proteins such as the 175 kD erythrocyte binding antigen (EBA175) and apical membrane antigen-1 (AMA-1) to the merozoite surface. Subsequently, interaction of EBA175 with glycophorin A (glyA), its receptor on erythrocytes, restores basal cytosolic calcium levels and triggers release of rhoptry proteins. Our results identify for the first time the external signals responsible for the sequential release of microneme and rhoptry proteins during erythrocyte invasion and provide a starting point for the dissection of signal transduction pathways involved in regulated exocytosis of these key apical organelles. Signaling pathway components involved in apical organelle discharge may serve as novel targets for drug development since inhibition of microneme and rhoptry secretion can block invasion and limit blood-stage parasite growth.     

NONLINEAR PROCESSES OF INTRACELLULAR CALCIUM SIGNALING AS A TARGET FOR THE INFLUENCE OF EXTREMELY LOW-FREQUENCY FIELDS

Theoretical analysis of peculiarities of reception of weak extremely low-frequency periodic signals by calcium-dependent intracellular regulatory systems was performed on the reduced “minimal” model for calcium oscillations suggested by Goldbeter et al. (Proc. Natl. Acad. Sci. USA 87, 1461–1465, 1990). The model considered the following calcium-dependent processes: the rise in intracellular free calcium concentration ([Ca²⁺]_i) due to calcium ionophore A23187 action on a cell, activation of the Ca²⁺ entry through calcium channels in the plasma membrane by the initial rise in [Ca²⁺]_i, and the Ca²⁺ release from intracellular stores by the calcium-induced calcium release mechanism. Calcium channels of plasma membrane were chosen as a target for the modulating signal and an additive noise influence in the model. An increase in [Ca²⁺]_i under the influence of the modulating signal was demonstrated to depend not only on the amplitude and frequency of this signal, but also on the phase of the signal with respect to a momentary chemical stimulation of the cell. Such an effect was found only at high strengths of chemical stimulation and with a particular sequence of delivery of the chemical and electromagnetic stimuli. An increase in noise intensity led to magnification of the mean level of [Ca²⁺]_i in a narrow frequency range by the mechanism of stochastic resonance. Under the influence of a modulating periodic signal, the gradual increase in strength of chemical stimulation induced a system transition from regular to chaotic behavior, and then to induced periodic oscillations. A boundary of the transition from chaotic to periodic oscillations corresponded to a “threshold” of sensitivity of calcium-dependent intracellular signaling systems on [Ca²⁺]_i to the influence of the modulating signal. Results of the theoretical analysis led us to conclude that the narrow-band response of a system to an external electromagnetic signal is determined purely by nonlinear properties of the system.     

Ca2+ dynamics in a population of smooth muscle cells: modeling the recruitment and synchronization

Many experimental studies have shown that arterial smooth muscle cells respond with cytosolic calcium rises to vasoconstrictor stimulation. A low vasoconstrictor concentration gives rise to asynchronous spikes in the calcium concentration in a few cells (asynchronous flashing). With a greater vasoconstrictor concentration, the number of smooth muscle cells responding in this way increases (recruitment) and calcium oscillations may appear. These oscillations may eventually synchronize and generate arterial contraction and vasomotion. We show that these phenomena of recruitment and synchronization naturally emerge from a model of a population of smooth muscle cells coupled through their gap junctions. The effects of electrical, calcium, and inositol 1,4,5-trisphosphate coupling are studied. A weak calcium coupling is crucial to obtain a synchronization of calcium oscillations and the minimal required calcium permeability is deduced. Moreover, we note that an electrical coupling can generate oscillations, but also has a desynchronizing effect. Inositol 1,4,5-trisphosphate diffusion does not play an important role to achieve synchronization. Our model is validated by published in vitro experiments obtained on rat mesenteric arterial segments.     

Equality of average and steady-state levels in some nonlinear models of biological oscillations

Nonlinear oscillatory systems, playing a major role in biology, do not exhibit harmonic oscillations. Therefore, one might assume that the average value of any of their oscillating variables is unequal to the steady-state value. For a number of mathematical models of calcium oscillations (e.g. the Somogyi–Stucki model and several models developed by Goldbeter and co-workers), the average value of the cytosolic calcium concentration (not, however, of the concentration in the intracellular store) does equal its value at the corresponding unstable steady state at the same parameter values. The average value for parameter values in the unstable region is even equal to the level at the stable steady state for other parameter values, which allow stability. This holds for all parameters except those involved in the net flux across the cell membrane. We compare these properties with a similar property of the Higgins–Selkov model of glycolytic oscillations and two-dimensional Lotka–Volterra equations. Here, we show that this equality property is critically dependent on the following conditions: There must exist a net flux across the model boundaries that is linearly dependent on the concentration variable for which the equality property holds plus an additive constant, while being independent of all others. A number of models satisfy these conditions or can be transformed such that they do so. We discuss our results in view of the question which advantages oscillations may have in biology. For example, the implications of the findings for the decoding of calcium oscillations are outlined. Moreover, we elucidate interrelations with metabolic control analysis. This paper is dedicated to the memory of the late Reinhart Heinrich, who was the academic teacher of S.S. and, to a great extent, also of M.M.     

Model of intercellular calcium oscillations in hepatocytes: synchronization of heterogeneous cells.

Hepatocytes respond with repetitive cytosolic calcium spikes to stimulation by vasopressin and noradrenalin. In the intact liver, calcium oscillations occur in a synchronized fashion as periodic waves across whole liver lobules, but the mechanism of intercellular coupling remains unclear. Recently, it has been shown that individual hepatocytes can have very different intrinsic oscillation frequencies but become phase-locked when coupled by gap junctions. We investigate the gap junction hypothesis for intercellular synchronization by means of a mathematical model. It is shown that junctional calcium fluxes are effective in synchronizing calcium oscillations in coupled hepatocytes. An experimentally testable estimate is given for the junctional coupling coefficient required; it mainly depends on the degree of heterogeneity between cells. Intercellular synchronization by junctional calcium diffusion may occur also in other cell types exhibiting calcium-activated calcium release through InsP(3) receptors, if the gap junctional coupling is strong enough and the InsP(3) receptors are sufficiently sensitized by InsP(3).